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By Charles Rhodes, P. Eng., Ph.D.
The following notes have been made from physical analysis and from Sensei Fisher's classes, comments and emails:
Any human motion can be expressed as a combination of Centre-of-Mass (CM) motion and limb motion with respect to the CM. A limb motion with respect to the CM is potentially much faster than CM motion due to the lower inertial mass of the limbs. Generally around a karate competitior's CM there is about a distance of about 1 m that the competitor can reach without significant CM motion. It is dangerous for a competitor to remain within the reach distance of his/her opponent because if the opponent initiates first the opponent will usually win due to the finite reaction time of the competitor. For this reason Sensei Fisher recommends that normally there should be at least one full step (~ 1 m) of the taller person between the front leg of a competitor and the front leg of his/her opponent so that if the opponent initiates first the competitor has time to appropriately react.
However, when the two fighters are that far apart the success of an attack is highly dependent on the ability of the initiating competitor to rapidly close the distance to his opponent. Much of kihon should be directed to minimizing the time required to close the distance to the opponent without prematurely alerting the opponent.
The purpose of kihon training (basic practice) is to develop flexibility, muscle memory, strength and timing for correct execution of basic techniques. These basic skills are maintained through daily practice.
It is important that individuals leading kihon practice structure the practice so that all the kihon movements make sense in a real kumite situation. Otherwise the training does not reflect reality and bad habits develop that are difficult to break.
An example of poor kihon practice is a triple punch from a stationary position, because it does not reflect a realistic kumite situation and hence develops incorrect muscle memory.
An example of much better kihon practice is a left jab-right reverse punch combination involving a step-slide to gain distance, a clockwise (CW) body rotation to power the left jab and a counter-clockwise (CCW) body rotation along with straightening of the right rear leg to power the right reverse punch. The jab is more of a push than a scoring technique but it serves to block the opponents vision so that the right reverse punch can score.
Every technique in karate-doh must be completed in < 0.5 seconds. This time frame accurately reflects reality. To train in a time frame of > 0.5 seconds is to deny reality.
In order to operate in this time frame both the upper and lower halves of the body must be loaded simultaneously. Sequential loading movements lead to > 0.5 second motions.
The body must always be straight and erect to enable turns that can be executed in < 0.5 seconds. Tuck the butt in and keep it tight. Keep the chin in to keep the spine vertical over its entire length. Do not look down.
Initially the legs are one forward, one back, knee motion planes and feet pointing at the target, knees slightly bent, rear heel almost touching the floor.
CENTER OF MOMENTUM (CM) DROP:
Every Fisher Shotokan technique is preceeded by a center of momentum (CM) drop to simultaneously load both the leg muscles and the upper body. The vertical drop causes a momentary sensation of weightlessness. There should be no horizontal motion during the vertical drop. This drop typically lasts ~ 0.15 seconds during which time a competitor must assess a threat and decide upon an appropriate response.
Forward motion is initiated by straightening the rear leg which compresses the rear foot. In this respect the heel of the rear leg must be firmly on the floor to develop maximum force and that the rear foot must be pointed in the direction of the desired CM motion to optimize engagement of the rear leg muscles. It is necessary through practice to develop a high degree of ankle flexibility to maximize the distance over which maximum force can be developed. Remember that the kinetic energy (KE) developed by this motion is given by:
KE = (force) X (distance)
so the larger the distance over which maximum force can be developed the better. If the initial posion is too low that distance will be too short. If the initial stance is too high it is impossible to maximally increase distance via step-slide or full-step motions.
Backward motion is initiated by hip rotation which pulls back the leading leg and shifts the weight onto the other leg. If a large angle turn is required arm motion is also used to increase angular momentum (rotational kinetic energy).
MAXIMUM DELIVERABLE ENERGY:
Deliverable energy is enhanced by engaging as many large muscles as possible. These muscles are typically the legs, hips, back and arms. In order to maximize energy transfer, the body must be rigid at the instant of impact. Hence every major muscle group must contract at the instant of impact.
Use proper body positioning to create the greatest speed possible. The largest muscles are in the legs for forward and back motion. In order to fully engage these muscles and hence achieve maximum speed it is important that the knee and foot of the loaded leg be pointed at the target at the Mid-point Fully Loaded Position (MFLP).
"Knee and foot pointed at the target" means that a vertical plane tangent to the inside edge of the heel and passing through the second big toe knuckle from the toe tip, if extended, will pass through the knee and through the target.
Each technique should contain an embedded retraction which creates a whipping action that maximizes energy delivery to the target.
The Critical Instant is the moment in time at which maximum rate of energy transfer is achieved. Anything that causes any involved joints to not reach their full rotation and/or that causes any involved muscles to not be at full contraction at the Critical Instant will diminish maximum rate of energy transfer. The usual cause of incomplete joint rotation and/or incomplete muscle contraction is improper distance from the target.
In order to achieve maximum energy delivery via a punch or kick to the opponent's body, at the Critical Instant the attacker's effective striking mass must be as large as possible implying that the attacker's entire body must be rigid at the Critical Instant.
As soon as a kihon technique is complete the body should be reset in preparation for the next technique. Failure to reset during kihon practice is a common error that leads to incorrect muscle memory.
Normal bi-pedal motion causes oscillating lateral CM motion. Advanced karateka sshould take this lateral motion into account.
During standing punches make sure that the outside edges of the feet are parallel. If the foot edges angle outwards the upper and lower halves of the body are not be unified. A common error during standing punch practice is failure to instantly reset.
1. A block should be executed in one continuous motion, not two counts.
2. Do not fall into a block. Drop low to load the legs and then drive in.
3. All blocks should be executed with a 50% - 50% weight distribution on the feet at the critical instant except for blocks in back stance (kukutsu-dachi).
4. Blocks need to be practiced in combination with simultaneous counter attacks.
5. In kukutsu-dachi the feet should be at greater than or equal to 90 degrees to enable lowering the front hip down.
During a back stance shuto block students should attempt to touch their shoulder blades together. This action will drive the striking or blocking hand forward faster and will provide a strong foundation for that hand when delivering energy to the opponent. The power for the shuto block is provided by the body trunk, not the striking arm. There is a hip turn into the block. Near the end of the turn the front or blocking side hip drops down with the pectoral muscles fusing the arm to the body.
To execute a forward side snap kick with the right leg step with the left foot in front of the right foot which causes CM motion to the right. The toes of the left foot should initially point at the target. Kick with the right leg. During the kick the left foot rotates 180 degrees about the ball of the left foot which maximizes kick reach and engages the left knee in the kick.
At the Midpoint Fully Loaded Position (MFLP) a vertical plane through the loaded leg should point at the target.
1. The normal kumite stance is both legs part loaded with approximately equal weight on each leg. Both feet should point toward the opponent. Any stance other than a normal kumite stance is only briefly used at the end of an attack or retreat motion when the initially loaded leg is fully unloaded. Generally one should return to the normal kumite stance as soon as possible.
2. The stance width should not be too narrow or too wide. A very narrow stance prevents rapid turns and rapid lateral movements. Too wide a stance presents the opponent too many targets.
3. Each karate technique involves: muscle loading to accumulate elastic potential energy, muscle contraction to convert chemical potential energy into kinetic energy, accumulation of kinetic energy via linear and angular momentum and impact with focus to transfer the maximum possible fraction of the available kinetic energy to the chosen target.
4. The body must always be straight and erect to enable turns that can be executed in less than 0.5 seconds. Tuck the butt in and keep it tight. Keep the chin in to keep the spine vertical through its entire length. Do not look down.
5. In Fisher Shotokan there is always a center of momentum (CM) drop to simultaneously load both the leg muscles and the upper body. The vertical drop causes a momentary sensation of weightlessness. There should be no horizontal motion during the vertical drop.
6. During loading there may be a rotation to face the target. Then there is a drive forward with a block/attack.
7. Deliverable energy is enhanced by engaging as many large muscles as possible. These muscles are typically the legs, hips, back and arms. In order to maximize energy transfer, the body must be made rigid at the Critical Instant of impact by engaging all muscles. At the Critical Instant the maximum rotation of all joints should be reached simultaneously with the maximum contraction of all of the major muscles.
8. Use proper body positioning to create the greatest speed possible. The largest muscles are in the legs for forward and back motion. In order to fully engage these muscles and hence achieve maximum speed in a linear attack at the mid-point fully loaded position (MFLP) the knee and foot of the loaded leg must be pointed at the target. In kumite having both feet pointed at the target brings into play either hand or foot. Contract every major muscle in the body beginning with the large muscles of the loaded leg and continuing to the smaller muscles of the impacting hand or foot.
9. "Knee and foot pointed at the target" means that a vertical plane through the knee and the long axis of the foot, if extended, will pass through the target.
10. A fully loaded leg permits movement in only one direction and contains elastic potential energy that soon decays. A fully unloaded leg contains no elastic potential energy and does not permit explosive discharge of chemical energy. A fully unloaded leg is not in a good position to initiate a movement.
11. Any stance that does not provide a 50%-50% weight distribution on the legs is the end of something and not a beginning. An exception to this rule is the 270 degree turn in kata, which rotates about the front leg. Due to the length of the turn and the 0.5 second one has to complete the technique, there is not enough time to move the CM back to a 50 -50 position at the completion of the front stance and then move the CM forward again over the pivoting heel of the loaded leg, which in this case, is the front leg of the front stance. Since the CM is already over the loaded leg of the next movement, there is no need to return to a 50%-50% position, because in kata, the next movement is known. Remember, when one leg unloads, the other leg automatically loads.
METHODS OF LINEAR MOVEMENT:
In kumite there is a stationary punch and there are four methods of linear movement. The initial position is 50%/50% in all cases. Each linear movement method can be executed either forward or backwards with either the left leg or right leg leading. These methods are compress-slide, step-slide, full step and any combination of these. Note that the back leg initiates forward motion and the hips initiate backward motion.
Of these methods of linear movement usually step-slide needs the most practice because it is seldom used outside the karate environment.
Compress-slide is used to achieve relatively short but very rapid distance changes from an opponent. Compress-slide is used going forward to attack an attack or going backward to absorb an attack and counter-attack at the end of the attack. Do a CM drop and compress with the leg opposite to the desired direction of motion while sliding the other foot. At the end of the technique the hips go square pulling up the rear foot.
In forward motion compress-slide the compressed back leg extends causing unloading of the front leg and causing the centre-of-mass (CM) and front leg to move forward one half step. As the back leg runs out of stored energy most of the weight briefly shifts onto the front leg while the back foot slides forward to catch up to the CM, at which point the weight distribution returns to 50%-50%.
Step-slide is used to achieve moderate but rapid distance changes from an opponent. Step-slide is used when greater distance changes are required than are possible with compress-slide. For forward motion compress the back leg to initiate forward motion. The vertical component of this force vector slightly lifts the CM. While the CM is in free fall quickly bring the back leg up under the CM and then discharge the back leg again while moving forward sliding the front leg. when the back leg is nearly discharged momentarily transfer the body weight onto the front leg and slide the back leg forward to its original position.
For backward motion initiate by spinning hips which pulls the front leg back to the fully loaded position and then discharge the energy in the front leg while sliding the rear foot backward.
Full-step is used to achieve a large distance change from an opponent. For forward motion always initiate off the back leg. In a forward motion full step the compressed back leg extends and unloads pushing the CM forward until the CM passes over and in front of the now fully loaded front leg. Then this loaded leg unloads to transfer at least 50% of the weight onto the new front leg. The point at which all the weight is directly over one leg is referred to as the Midpoint Fully Loaded Position (MFLP).
For rearward motion always initiate with the hips.
A full step like motion can also be used to change feet in place.
In kumite, shia and uncontrolled situations, jabs are never done from a stationary position. Jabs are always moving forward or, in reverse with a compress-slide as a minimum. A stationary jab is done only during kihon practice. Also, a jab always starts and finishes with the punching hand on the forward hip. For a jab to be complete, or, any punch for that matter, it goes out from and then goes all the way back to the hip in order for the punch to be complete. Left hip for the left hand punch, right hip for the right hand punch. At impact the punching arm must be nearly straight. The speed the punch achieves during discharge of the attacker's CM kinetic energy is directly proportional to the speed which the loaded leg emptys.
Remember that any front hand technique's fully loaded position (MPFLP) has the shoulders square to the target. In this position it can be said that the upper body is fully loaded. Simultaneously, the loaded leg reaches it's fully loaded position (MPFLP). And now it can be said that the lower half of the body is fully loaded. For a technique to be fully loaded, which is the primary objective in the study of Karate-doh, both the upper and the lower halfs of the body must be fully loaded when calculating the total energy output. The < 1/2 second time scale requires simultaneous loading of the upper and lower halfs of the body. Included of course, in a full loading is the proper positioning of the arms and legs during both the loading phase and the unloading phase. That in essence is what the study of Karate is, an understanding of how to load and unload the body for a given purpose.
In any punching technique the motion ot the pulling hand is essential both to provide angular momentum and to drive the opposite arm forward via the rigid shoulders.
In Fisher Shotokan there can be no talk of any technique which does not include the emptying of the loaded leg. This includes a jab. Even done in a stationary position, as is done in kihon practice, the knee of the loaded leg always empty's symultaniously with the elbow of the striking hand.
TURNS – GENERAL COMMENTS:
1. Turns should move the defender's CM off the line of attack.
2. All turns are executed backwards. If the pivot point is the ball or heel of the right foot the turn is Counter Clock Wise (CCW). If the pivot point is the ball or heel of the left foot the turn is Clock Wise (CW). The human skeleton is not well adapted to performing forward high energy turns.
3. The rate of a turn is known as its angular velocity W. The angular momentum L of a rotating real body is given by the formula:
L = I W
I = moment of inertia
W = angular velocity vector usually expressed in radians / second.
When not turning W = 0 and hence L = 0.
4. The parameter I is related to the mass distribution about the rotation axis.
dI = dM R^2
For a body rotating about a vertical axis I is minimized by keeping the body and limbs as close to the vertical axis of rotation and as upright as possible.
5. The angular momentum L is a vector quantity that for a rigid body does not change during free rotation. Thus during free rotation for a rigid body:
L = Ia Wa = Ib Wb
where subscripts a and b indicate two successive times.
However, by reducing I during a rotation W can be increased. Moreover pullinng the arms and legs toward the rotation axis during a rotation requires work which further increase the angular momentum (rotional kinetic energy).
6. In Fisher Shotokan, prior to commencement of free rotation the initial angular momentum L is made as large as possible with a large initial moment of inertia Ia. Then after free rotation commences I is reduced from Ia to Ib in order to force the angular velocity W to increase from Wa to Wb. This process increases both the angular velocity and the kinetic energy.
7. The angular momentum L of a rigid body can only be changed by application of torque. Torque is a twisting action that can be viewed as an axial flow of angular momentum. However, realizing a large flow of vertical axis angular momentum into or out of a human body requires that both feet be firmly on the ground.
8. Each Fisher Shotokan turn has three sequential torque phases. During the first phase both feet are on the ground and angular momentum flows into the body. During the second phase there is free rotation about the ball or heel of one foot and the amount of angular momentum within the body is increased by pulling the arms and legs toward the axis of rotation. During the second phase the moment of inertia is minimized. The unloaded leg must not land before the critical instant of impact. At the critical instant of impact both feet are again on the ground and angular momentum (kinetic energy) flows out of the body.
9. In Fisher Shotokan the angular velocity (rate of turn) during free rotation is largely limited by the amount of angular momentum that flows into the body during the short initial time interval while both feet are still in firm contact with the ground and by the practical available change in the karateka's moment of inertia.
10. For a rigid body angular momentum is the time integral of torque.
11. The muscles that can cause vertical axis torque are the hips and shoulders acting against the upper body moment of inertia while both feet are firmly on the ground. For any initial moment of inertia Ia, the larger the angular velocity Wa that can be achieved while both feet are on the ground, the larger the amount of angular momentum that is absorbed by the upper body and hence the faster the turn. Thus a fundamental objective of Fisher Shotokan physical training is development of maximum hip and shoulder angular acceleration about the vertical body axis while the arms are outstretched and both feet are on the ground.
12. Note that while the body's angular momentum is changing the torque is zero at the top of the head, is moderate at the shoulders and is very large at the hips.
13. The average angular velocity during a turn is maximized by generating angular momentum L at a large initial value of moment of inertia I = Ia and then reducing the moment of inertia to I = Ib as soon as free rotation commences. Conservation of angular momentum dictates that the angular velocity after the moment of inertia reduction will be:
|Wb| = (Ia / Ib) |Wa|.
Since if Ia > Ib, then: Wb > Wa
14. Turns can be divided into three general categories.
15. First Category: The first category of turns consists of limited angle turns that can be achieved by rotating the hips which causes the legs, feet and the body trunk to rotate in position with only a slight lateral Center of Momentum (CM) movement. The trunk rotates about its vertical axis through the trunk CM. Each leg rotates about its own long axis. This turn is very fast due to a small moment of inertia but the maximum angle of rotation is limited to less than 90 degrees.
16. Second Category: The second category of turn consists of turns where one foot moves laterally moving the body trunk CM and the body trunk rotates about a vertical axis through its CM. Each leg rotates about its long axis. The loaded foot rotates in place on its ball and the other foot follows the hips. An example is the first 90 degree turn in the kata Heian Shodan. The moving foot should remain close to the CM vertical axis until after the body has turned. Only then should the moving foot move out to its final position.
A common error during back and forth kihon practice is executing 180 degree turns by moving the rear foot directly from its initial position to its final position without moving it close to the CM vertical axis before turning as in a Third Category turn. This error develops incorrect muscle memory and sacrifices power and response options.
17. Third Category: The third category of turn consists of turns of any angle that are achieved by shifting the body CM over one leg and rotating on that leg. Within this third category there are subcategories for rotations on the heel of the foot, the ball of the foot and the flat of the foot. There are also further subcategories for rotations that pivot about the front leg or back leg. An example of the third category of turn is the 270 degree turn in Heian Shodan. In kata the sequence of motions is known, which allows anticipation as to which leg to preferentially load during a turn. There are numerous third category turns where the distance to the opponent or lateral shift requirements may dictate whether the turn is made on the ball or the heel of the loaded leg. There are back stance turns that are made on the flat of the rear foot. Due to the large number of types and variations of turns, in each case the Fisher Shotokan concept of maximizing deliverable kinetic energy in minimum elapsed time must be applied to dictate the detail of the turn.
An important advantage of third category turns is that they can be used to provide lateral movement to either avoid an attack or to properly position the body for a counter attack.
A special case is the 180 degree turn that is used to change direction in most back and forth kihon practice. In that turn the moving leg should move in under the CM before the body turns and then move out again after the body turns. Most karateka execute this turn as a simple second category turn, but that is a mistake. A third category turn provides more defense and counter attack options.
Another important special case is avoiding a high energy punch or kick to the body by explosively rotating the body trunk immediately prior to sliding forward or stepping backward. This explosive trunk rotation involves a "hip flip" (a partial hip spin) and is a distinguishing element of both advanced JKA Shotokan and Fisher Shotokan. Mastering the backward "hip flip" is easier if there is a serious element of fear of being hit during kihon kumite.
LIMITED ANGLE TURNS EXECUTED BY ROTATING THE FEET:
1. Generally a Fisher Method 90 degree turn such as the first movement of Heian Shodan involves a backwards rotation on the ball of one foot with the other leg relaxed and following the hips. This footwork causes the body to rotate about the CM with a relatively small moment of inertia. There is only a small translational CM motion during the turn. The footwork enables a very fast turn to a fully loaded position.
2. The Fisher Method of executing limited angle turns allows for creation of a greater amount of translational kinetic energy immediately after the turn than the JKA Method. In the Fisher Method a 90 degree turn involves lowering the center of mass (CM) while rotating on the ball of the foot of the loaded leg. The time required to implement this turn cannot be shorter that time required to lower the CM, which time is set by the CM drop distance and the acceleration of gravity. The CM drop transfers gravitational potential energy to the elastic muscles of the legs, so the CM drop is crucial to maximizing the available potential energy for a followup strike.
3.Turning on both heels at the start of a kata does not fully load the loaded leg. Turning on both heels will not allow a full CM drop and will leave the CM insufficiently forward of the rear loaded leg and insufficiently behind the front loaded leg. Turning on both heels also takes longer to accomplish than turning on one ball because of difficulty generating torque.
DIRECTION REVERSAL 180 DEGREE KIHON TURNS:
1. Consider the 180 degree turn usually used to change direction in kihon practice. This is the maximum angle turn that can be executed by rotating the feet. Assume that at turn initiation the right foot is forward (leading) and the turn is CCW.
2. Glance over the left shoulder. Execute a CM drop together with a trailing (left) leg compression that shifts the CM toward the ball of the lead (right) foot. The unloaded trailing (left) leg moves in toward the right leg. Simultaneously the right hand punches backwards under the left arm. The left fist goes to gedan block MPFLP position at the right shoulder. Simultaneously the right leg rotates 180 degrees about the vertical axis through the ball of the right foot. During this rotation the body CM shifts laterally and the trailing (left) leg rotates about its long axis. The feet move with the legs.
3. The trailing (left) leg rotates in the air and then moves back out toward the opponent to become the new lead leg. It may finish its rotation on the heel of this foot. The loaded right leg and left arm unload to gedan block/attack toward the opponent. During this turn the position of the new leading (left) foot laterally shifts by two shoulder widths.
4. A frequently encountered error is to execute this turn without moving the trailing (left) foot in towards the CM and then out again. A problem with this error is that it limits a karateka's options if he/she is attacked from behind. By habitually drawing the trailing leg toward the CM the karateka reduces the turn moment of inertia and has many more response options to an attack from the rear.
5. The Fisher Method of executing a 180 degree kihon turn allows for creation of a greater amount of translational kinetic energy immediately after the turn than the JKA Method. In the Fisher Method a kihon turn involves lowering the center of mass (CM) while rotating on the ball of one foot. The time required to implement this turn cannot be shorter that time required to lower the CM, which time is set by the CM drop distance and the acceleration of gravity. The CM drop transfers gravitational potential energy to the elastic muscles of the loaded leg, so the CM drop is crucial to maximizing the available potential energy.
6. If the opponent is too close the Fisher Method of executing a kihon turn can be almost instantaneously modified to a backwards rotation on the balls of both feet to gain a small amount of distance.
7. After a kihon turn do not fall into a block. Use the energy contained in the loaded (right) leg to drive toward the opponent.
LARGE ANGLE TURNS:
1. Large angle turns involve moving the CM over one leg, a rotation about that leg and moving the CM out again.
2. Large angle turns, like all karate techniques, have two phases, a loading phase and an unloading phase that must be executed continuously.
3. Such turns involve shifting the CM close to the axis of rotation at the start of the turn and then shifting the CM away from the axis of rotation at the end of the turn.
4. The hips provide the majority of angular momentum for any given turn. Swinging the arms provides additional angular momentum for any given turn.
5. In most large angle turns about one leg, the loaded leg pivots on the heel whereas small angle turns are done on the ball. However, the distance to the target supersedes all other considerations and may require the loaded leg to rotate on the ball of the foot.
6. Rotations in a back stance such as in Heian Nidan usually occur about the center of the foot but may be affected by ankle flexibility.
7. Each Fisher Shotokan turn about one leg involves a multiplicity of important sequential elements.
a) Vertical alignment of the spine axis through the CMs of the pelvis, trunk and head. The ass and chin are tucked in to achieve this vertical alignment at both the bottom and top of the spine;
b) Accumulation of potential energy in muscles via a CM drop;
c) Accumulation of angular momentum at a large moment of inertia via extended arm, extended leg and powerful hip rotation movements;
d) Shifting the CM directly over the leg about which rotation will take place;
e) Reduction in the moment of inertia about the vertical pivot axis to maximize angular velocity and increase kinetic energy. The spine axis, arms and unloaded leg are all drawn close to the pivot axis;
f) Rotation about the vertical pivot axis to the desired angle;
g) Enlargement of the moment of inertia to slow rotation;
h) Shift in CM away from the pivot axis towards the target;
i) Transfer of kinetic energy to the target;
j) Use of the unloaded foot as a brake to discharge angular momentum and stop the rotation.
k) The foot of the unloaded leg must not come to rest until after the loaded leg is fully unloaded.
1. If a 180 or 270 degree turn is done on the ball of the foot of the loaded leg, during the rotation the loaded leg is not fully loaded. After the rotation is complete it is necessary to press the heel of the loaded leg into the ground to fully load that leg. If at the end of the rotation the foot of the loaded leg does not point at the target the loaded leg cannot not be fully loaded.
2. To minimize turn resistance when rotating on the heel of a foot, curl up the toes of that foot to prevent the ball of that foot dragging and to ensure that the ankle of that foot is fully loaded.
3. The detail of 180 degree and 270 degree large angle turns rotating about the heel of the loaded leg is set out on the web pages titled Turn Dynamics, Heian Shodan 2nd Turn and Heian Shodan 4th Turn.
This web page last updated May 27, 2017.
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